Transcript Lesson 5.3A

Geometric
GeometricSequences
Sequencesand
andSeries
Series
• How do we find the terms of an
geometric sequence, including
geometric means?
• How do we find the sums of geometric
series?
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Algebra 2Algebra 2
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Geometric Sequences and Series
Serena Williams was the winner out of 128 players
who began the 2003 Wimbledon Ladies’ Singles
Championship. After each match, the winner
continues to the next round and the loser is
eliminated from the tournament. This means that
after each round only half of the players remain.
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Geometric Sequences and Series
The number of players remaining after each round
can be modeled by a geometric sequence. In a
geometric sequence, the ratio of successive
terms is a constant called the common ratio
r (r ≠ 1) . For the players remaining, r is .
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Geometric Sequences and Series
Recall that exponential
functions have a common
ratio. When you graph the
ordered pairs (n, an) of a
geometric sequence, the
points lie on an exponential
curve as shown. Thus, you
can think of a geometric
sequence as an exponential
function with sequential
natural numbers as the
domain.
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Geometric Sequences and Series
Identifying Geometric Sequences
Determine whether the sequence could be geometric or
arithmetic. If possible, find the common ratio or difference.
1. 100, 93, 86, 79, . . .
7 7 7
Diff.
93 86 79
,
,
100 93 86
Ratio
It could be arithmetic,
with d = –7.
2. 180, 90, 60, 15, . . .
Diff.
 90  30  45
Ratio
90 60 15
,
,
180 90 60
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It is neither.
Geometric Sequences and Series
Identifying Geometric Sequences
Determine whether the sequence could be geometric or
arithmetic. If possible, find the common ratio or difference.
3. 5, 1, 0.2, 0.04, . . .
Diff.
 4  0.8  0.16
Ratio
1 .2 .04
,
,
5 1 .2
It could be geometric,
with r = 0.2.
4. 1.7, 1.3, 0.9, 0.5, . . .
Diff.
Ratio
 .4  .4  .4
1.3 0.9 0.5
,
,
1 . 7 1.3 0 . 9
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It could be arithmetic,
with d = –0.4.
Geometric Sequences and Series
General Rule for Geometric Sequence
The nth term of an geometric sequence is
an  a1 r 
n 1
where a1 is the first term and r is the common ratio.
Recursive Rule for Geometric Sequence
To state a recursive rule of an geometric sequence, state
the first term and the rule in the following form:
a1  ____; an  r  an1
where a1 is the first term and r is the common ratio.
Holt McDougal Algebra 2
Geometric Sequences and Series
Finding the nth Term Given a Geometric Sequence
Write a rule for the nth term of the geometric sequence.
Then find a7.
5. 4,
4 12, 36, 108, 324, 
12
r
4
3
an  a1 r
a7  43
n 1
7 1
a7  2916
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 43  4729
6
Geometric Sequences and Series
Finding the nth Term Given a Geometric Sequence
Write a rule for the nth term of the geometric sequence.
Then find a7.
3
3 3
3 3
6.
,  ,
,  ,
,
4
8 16
32 64
 3 / 8  11 
r
  
3 / 4  22 
an  a1 r
3 1
a7    
4 2
Holt McDougal Algebra 2
n 1
7 1
3

256
Geometric Sequences and Series
Finding the nth Term Given a Geometric Sequence
Write a rule for the nth term of the geometric sequence.
Then find a7.
7. 00..001
001, 0.01, 0.1, 1, 10, 
0.01
r
10
10
0.001
n 1
an  a1 r
a7  0.00110
7 1
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 1000
Geometric Sequences and Series
Finding the nth Term Given Two terms of a Sequence
Write a rule for the nth term of the geometric sequence.
Then find a8.
51
5
1
3
5
4
5 3
1
5
3
2
1
8. a  36, a  324
a  a r
324  36r
36
36
a  a r
324  a  3
324  81a
81
a1  4
r 9
r 3
2
a8  4 3
81
an  4  3
n 1
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81
 8748 or  8748
Geometric Sequences and Series
Finding the nth Term Given Two terms of a Sequence
Write a rule for the nth term of the geometric sequence.
Then find a8.
51
5
1
4
5
4
5 4
1
5
4
1
1
9. a  8, a  40
a  a r
 40  8r
8
8
r 5
625
625
a1  0.064
an   0.0645
n 1
Holt McDougal Algebra 2
a  a r
 40  a 5
 40  625a
a8  0.0645
81
 5000
Geometric Sequences and Series
Writing a Recursive Rule for an Geometric Sequence
Write the recursive rule for the nth term of the sequence.
10. 625, 125, 25, 5, 1, 
125 11
r

625 55
a1  625
an  r  an1
Holt McDougal Algebra 2
Geometric Sequences and Series
Writing a Recursive Rule for an Geometric Sequence
Write the recursive rule for the nth term of the sequence.
11. 3,  9, 27,  81, 243, 
9
r
 33
3
a1  3
an  r  an1
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Geometric Sequences and Series
Lesson 5.3 Practice A
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